Redundant fast fourier transform data handling computer

ABSTRACT

Special purpose data handling equipment is utilized with a fast Fourier transform algorithm computer. The equipment buffers input data such that each block of input data includes data from the immediately preceding and following data blocks. This input redundancy is useful when input data is processed through a data window which attenuates data at the beginning and end of the data block, since, when this data is transformed and subsequently inversely transformed, simple addition of the output data points removes the effect of the attenuating window.

United States Patent [1 Schmidt 1 1 REDUNDANT FAST FOURIER TRANSFORMDATA HANDLING COMPUTER [75] Inventor: Ralph Otto Schmidt, Santa Ana,

Calif.

[73] Assignee: Interstate Electronics Corporation,

Anaheim, Calif.

[22] Filed: May 18, 1973 [21] Appl. No: 361,533

Related US. Application Data [63] Continuation of Ser. No. 182,685,Sept. 22, 1971,

abandoned.

[52] US. Cl. 235/152; 324/77 B [51] Int. Cl. G06l' l5/34 [58] Field ofSearch 235/152, 156; 324/77 R,

324/77 B, 77 D; 179/1 SA [56] References Cited UNlTED STATES PATENTS11/1970 Klund 235/152 X OTHER PUBLICATIONS A. V. Oppenheim, SpeechSpectrograms Using the FFT" IEEE Spectrum Aug. 1970, pp. 57-62.

[ May 13, 1975 Primary ExaminerEugene G. Botz Assistant Examiner-DavidH. Malzahn Attorney, Agent, or Firm-Knobbe, Martens, Olson. Hubbard &Bear [57] ABSTRACT Special purpose data handling equipment is utilizedwith a fast Fourier transform algorithm computer. The equipment buffersinput data such that each block of input data includes data from theimmediately preceding and following data blocks. This input redundancyis useful when input data is processed through a data window whichattenuates data at the beginning and end of the data block, since, whenthis data is transformed and subsequently inversely transformed, simpleaddition of the output data points removes the cf fect of theattenuating window.

8 Claims, 12 Drawing Figures OUTPUT TIME (DEFF/C/E/VTS REDUNDANT FASTFOURIER TRANSFORM DATA HANDLING COMPUTER RELATED APPLICATION This is acontinuation of application Ser. No 182,685, Sept. 22, 1971, nowabandoned.

BACKGROUND OF THE INVENTION A. Fourier Analysis The frequency domaincharacteristics of a time do main waveform provide a powerful analyticaltool in a variety of technical disciplines. The Fourier analysis of agiven periodic function represents it as a sum of a number, usuallyinfinite, of simple harmonic compo nentsv Because the response ofalinear dynamic system to a simple harmonic input is usually easy toobtain, the response to an arbitrary periodic input can be obtained fromits Fourier analysis. Likewise. the field of spectrum analysis, byproviding a frequency domain representation of a waveform, facilitatesidentification of unknnown waveforms.

B. The Fourier Transform The Fourier transform, which is defined asfollows:

WHERE:

A(fJ=THE FREQUENCY DOMAIN FUNCTION X (l)=THE TIME DOMAIN FUNCTION is acommon mathematical tool for deriving the fre quency domain functionfrom a given time domain function and vice versa, i.e., given aparticular mathematical function which defines the amplitude variationsof a waveform with time, equation (1) above may be utilized to determinethe various frequency components of the time domain function. TheFourier trans form depends upon the underlying realization that anyperiodic amplitude function in the time domain may be constructed bysuperimposing a variety of sine and cosine functions in the time domain.each of these functions having a predetermined amplitude.

Early attempts to mechanize the Fourier transform utilized analogtechniques which consisted primarily of the application of a time domainwaveform to a series of bandpass filters, such that the response fromeach of the filters as indicative of the amplitude of the frequencydomain components of the time domain function within a given frequencyband width. Because of the requirement for a large number of filters, inorder to cover a broad over-all pass band while maintaining the passbands of the individual filters reasonably narrow, such techniques werenotable for their high equip ment cost. These costs were reduced throughthe use of a number of analog techniques, including the reiteration of agiven time sample through a single bandpass filter, while changing thecenter frequency of the filter for each iteration, thereby allowing onefilter to selectively sample for a plurality of frequency components.This reiteration, however, required additional time,

and the input waveform could be sampled only once during each of therelatively long reiteration periods.

C. The Discrete Fourier Transform Early attempts to use digitaltechniques for determining the frequency domain function correspondingto a time domain waveform utilized the discrete Fourier transform whichallows frequency domain components to be derived from a set of periodicdiscrete amplitude samples of an input time domain waveform. Thesetechniques were likewise limited to a periodic sampling of the inputwaveform. but the speed of digital calculation allowed a reduction ofthe sampling period from that required by analog techniques. Theanalogous dis crete Fourier transform pair that applies to sampledversions of the functions given in equations l and (2) can be written inthe following form:

1w amp; A/re 7 WHERE:

A(r)=THE r' COEFFICIENT OF THE FRE- QUENCY DOMAIN FUNCTION X(k)=THE kCOEFFICIENT OF THE TIME DO- MAIN FUNCTION Since the discrete Fouriertransform samples must be taken during a predetermined time period, theinput time domain waveform is assumed to be periodic in the time domain,and to have a period which is equal to the sampling time, i.e., theerrors introduced by truncating the time domain waveform are assumed tobe negligi ble. In fact, these errors are often not negligible, andnumerous techniques have been utilized to diminish the effect of thetime domain truncation, such as the superimposition upon the time domainsamples of a cosinesquared function. which is commonly termed hanning",in order to smooth the transition into and out of the time samplingperiod.

In order to accomplish the discrete Fourier transform of equation (3) orthe inverse discrete Fourier transform of equation (4), the equations(3) and (4) require N computations, each computation including amultiplication and an addition. For a reasonably precise transform, suchas one where N=2U48, the computations exceed 4,000,000, and thereforeeither an excessive amount of hardware is necessary, or an unreason ablylong time is required to complete the computation if the same hardwarehandles successive computations.

D. The Fast Fourier Transform Algorithm A fast Fourier transformalgorithm was derived by I. W. Cooley and J. W. Tukey and initiallypublished in Malhematics of Computation, volume 19, pages 297 to 301April, I965. This algorithm recognizes the similarities in a number ofthe N multiply and add computations of the discrete Fourier transform,and utilizes these similarities to reduce the total number ofcomputations required to calculate the discrete Fourier trans form. Byusing the fast Fourier transform algorithm outlined by J. W. Cooley and.I. W. Tukey, the computations are reduced from N operations to N/(2 logN complex multiplications, N/(2) log N complex additions, and N/(Z) logN complex subtractions. For N=2.048, for example. this represents acomputational reduction of more than 400 to I over the direct discreteFourier transform computations.

SUMMARY OF THE INVENTION The present invention involves special purposedata handling equipment which is utilized when attenuated input datawindows are used with a fast Fourier transform computer to change thecharacteristics of the filters sinthesized by the computer. Thisequipment, by redundantly inputting to the computer certain input data,allows the effects of the attenuated input data window to be easilyremoved.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. I is a computational flow chartfor the normal ordered Fast Fourier Transform algorithm where N=8;

FIG. 2 is a computational flow chart for the normal ordered Fast FourierTransform algorithm where N=32;

FIG, 3 is a computational flow chart for the reverse ordered FastFourier algorithm where N=8;

FIGS. 40 through 4d are plots showing attenuation of various datasamples with time using a cosine squared formation:

FIGS. 50 through 5d are plots showing attenuation of various datasamples with time using a trapezoidal attenuation formation; and

FIG. 6 is a block diagram of a data handling system for redundantlyhandling data in accordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT A. The Fast FourierTransform Algorithm In 1965 a paper was published by .I. W. Cooley andJ. Wv Tukey which described in algorithm for rapidly calculating thediscrete Fourier transform. This algorithm was adaptable to digitalcomputation of the trans form. and reduced the discrete Fouriertransform of 1 f0. A/amdfiii 5. W=e(i21r/N) equations (3) and (4)become:

IV! Mk5; ra /4W 0 IV! Z Mk2 Ami w?" WHERE:

r=O,l, N-l

k=0,l N-l It should be noted that the discrete Fourier transformexpression is complex, and that it is therefore possible to transformtwo series of time samples simultaneously, treating one of these seriesas the real part and one of the series as the imaginary part of thecomplex expression. It is likewise possible to transform 2N samples withthe expression defined in equations (3) and (4) by handling the evennumbered samples as the real part of the complex transform and the oddnumbered samples as the imaginary portion of the complex transform.Thus. for example, a 2048 point transform can be per formed as a 1024point (N=l024) complex transform.

When r and k are defined using binary notation. and when, for example, Nequals 8, r and k can be defined follows:

8. r-=4r 2r r 9. k=4k 2k k WHERE;

r=O, I, 7 F0, 1, 7 2: la ow 2\ 17 0 I This definition will, of course,change as the value of N changes. For example, for N=64, r=32r +l 6r +8r+4r +2r,+r

Using this notation for N=8, equation (6) becomes:

equations (3) and (4) to a series of sums. The derivas5 SINCE:

tion of this algorithm is briefly described as follows: By defining:

Mam 2 W WIZ'I'YLZWZWI' equation 10) may be written as follows:

The individual terms of equation (1 l can be written in timecoefficients which are 4 samples, or N/Z, apart are the following form:summed. The results of the first summation appear as 12. W": 'l 'o ":=[W*"l "a]W"o"2 binary coefficients along the line X in FIG. 1. Succesl3.W"2 1"o "|=[W":l]W i o l sive summations X and X in turn, require k, andk,,, l4. W 2 '1 *'0" =W: '1 0"0 respectively, to change from 0 to l, sothat coefficients Note, however, that due to the definition of W inequaresulting from the previous summation which are 2 cotion (5),efficients, or N/4; and one coefficient, or NIB, apart,

l respectively, are summed. The results of the summa- Thus, thebracketed portions ofequation (l2) and l 3) tions X and X, appear asbinary coefficients along the can be replaced by a one. It should benoted that, as N [0 lines X and X in FIG. 1. Regardless of the value ofN,

changes, equation will change, but that a number of the first summationwill always involve the combination terms of the equations will equal lregardless of the of samples which are N/2 samples apart, the secondvalue of N. Therefore, equation (l0) can be written in summation N/4apart, and so on, until enough passes th f l ing fOYmI 15 have beencompleted so that adjacent coefficients are i I J 1 )k I {2, I" 2k 4*Z'pr 1e. Momma-{Z 5 E wraith/ m W w o f0 f0 In this form, each of thesummations can be sepacombined. For example, FIG. 2 shows that for N=32,rately performed, and only the results of the last sumsamples which arel6, 8,4, 2 and l apart are combined,

matlon must be saved. Thus, equation 10) can be perso that fivesuccessive summations are requlred for formed as the series ofsuccessive summations shown N=32. As shown in FIG. I, the binarycoefficients of below: A(r) are not reordered, as shown in equation(20);

rather, the output frequency domain coefficients in 1 FIG. 1 occur asordered in equations (19), i.e. in reverse binary orderfrom A to A or,in binary notation, A2; x/ ml ljJll from 000 to [11. Similarly, theoutput coefficients in FIG. 2 occur in reverse binary order from 00000to lllll. Reverse binary order is defined as the order 2/; iv g achievedwhen the order of the binary digits of words fa A2 2 W/ a) I in normalsequential order is reversed, i.e., 110 becomes 01 I, so that for N=8,the third coefficient appears where the sixth coefficient would appearif the ff e 2 X l/j/ order were normal, or for N=32, 0| I00 (12) becomes00110 (6).

The w notation in FIGS. 1 and 2 denotes the power to which W is raisedfor each of the calculations inf U, G) 0 volved in the summations X X,,and X Thus, in FIG.

1, it can be seen that for the production of the first binarycoefficient of X the complex coefficient X, is combined with the complexcoefficient X, after the cowhsre Xa IS first Summation Xb Is the secondefficient x, is multiplied w". similarly, for the produc matron and X0is the third summation. These three suclion 0f the frequency d i ffi i tA h hi d cesswe summations P Serve compute the coefficient of X iscombined with the fourth coeffivalue ofA (r) when N is equal to 8. In asimilar manner, dent f X0 multipned by wt h ld b noted h greater numbersof successive summations Wm each time a pair of coefficients is combinedto produce P the Value of larger Values For exam a new pair ofcoefficients, the combination requires a P i when N=1024i ten Successivesummations will complex multiplication, a complex addition, and a i theValue Of complex subtraction. For example, in the combination Thesummations Show" in equations 17 through 19 0 of input coefficients Xand X, to produce the first and are diagrammed in 1 wherein through 1fifth coefficients of x,,, x is combined with x, multinote periodicconsecutive time sample pairs correli d b either I or W, Fr m e uation(5), when sponding to the amplitude of a function to be trans- N=8, W ith ti f W 50 that it is necessary formed. The first time sample Of eachpair is treated as to X4 X w only nce and then add and sub. the realpart of the complex coefficient while the sect t th product f to producethe two co ffi- 0nd Sample is treated as the imaginary P Thesecoefcients of X This negative relationship holds true for ficientscorrespond to binary notation coefficients 000 w =-W, W W and W=W whenN=8. Simithrough lll, as shown in FIG. 1. The first summation, l rly, inFIG. 2, when N=32, W=W, W=W", Xa, requires it, to change from O to I, sothat periodic W==W Section B. Reverse Binary Order Algorithm FlG. 3shows the computational flow diagram for the algorithm explained abovewherein the input samples are in inverse binary order rather than thenormal order.

It can be seen that when the input samples are in reverse binary order,the first summation or pass of the algorithm requiring samples which areseparated by four coefficients when N=8, appear adjacent one another.Therefore, the first pass in the algorithm utilizing reverse binaryordered input coefficients, combines ad jacent samples. Similarly, thesecond pass combines samples which are separated by two coefficients andthe last pass to produce X,. combines samples which are 'fourcoefficients apart, these being adjacent coefficients in a normalordered pattern. The algorithm of FIGS. 1 and 3 are identical, and theactual calculations which are carried out combine the same coefficients.The only thing which has been changed is the order in which the inputcoefficients appear. It will be noted from FIG. 3 that when the inputdata samples are in re verse binary order, the output appears in normalorder. For this reason, input data is often transformed, mathematicalequations are conducted on the reverseordered frequency coefficients andthe newly computed frequency coefficients are inversely transformed tocreate new time-domain coefficients showing the effect of the frequencydomain calculations. It can be seen from the similarity of equations 3and 4 above that the format required for the application of the inversetransform is identical to that for the forward transform. Thecalculation of the inverse transform on the reverse binary orderedfrequency coefficients therefore results in time domain coefficients innormal binary order.

As noted above, in order to enhance the characteristics of the filterswhich are simulated in the Fourier transform and to simultaneouslydecrease the production of unwanted frequency domain characteristics dueto the sampling of data during a discrete time window rather than overan infinite period of time, it has been found advantageous to attenuatedata at the initiation and conclusion of each time window by variousformulae. One such formula, and the one utilized in the preferredembodiment of the present invention, is the square of a cosine having aperiod equal to the time window. This cosine squared function issuperimposed on the input data samples, as shown in FIGS. 40 through 4c,so that samples in the center of the window are not attenuated, butsamples are attenuated to a greater degree toward the extremity of thetime window.

Although such an attenuation scheme, often referred to as hanning,produces the desired results indicated above, it will be readilyunderstood that when data is transformed into its frequencycoefficients, and later inversely transformed to produce time domaincoefficients, these calculated time domain coefficients will include theeffects of the initial attenuation. Therefore, it has been the custom tomultiply such calculated time domain coefficients by the inverse of theattenuating waveform, thereby creating the opposite effect of theinitial attenuation and resulting in meaningful time domain coefficientsThe present invention alleviates the necessity of mul tiplyingcalculated time domain coefficients by an inverse waveform through theuse of special data handling procedures which result in the calculationof meaningful time domain coefficients by the inverse fast Fouriertransform algorithm itself.

C. Data Handling Method The redundant data handling technique of thepres ent invention is best described in reference to FIG. 4 which showsa plot of one type of attenuation curves which may be applied to inputtime domain samples prior to Fourier transform analysis. In the exampleshown in FIG. 4, each data set to be transformed is attenuated using anidentical attenuation curve, the attenuation curve 12 being a banningcurve, or cosine squared curve, plotted against time. This attenuationcurve is applied to a complete block of input time domain samples whichare to be used as a complete data set for the calculation of a FastFourier Transform. Thus, for example, if an eight point transform is tobe conducted, the time T-() through T-2 in FIG. 4a will be the timerequired for eight real time samples to be measured from the inputwaveform which is being sampled and applied to the input of the FastFourier analysis computer. The first and last of these samples are attenuated to a greater extent than is a sample in the middle of the data setso that a cosine squared function is superimposed upon the data set.When such attenuation was performed in the past, it was common toutilize, as the next data set, the input coefficients which appearbetween times T2 and T4, again superimposing upon these time samples anattenuation curve. The present invention. however, stores the latterhalf of the time samples in the first data set; that is, those timesamples which were monitored between times T-l and T-2, and utilizesthese data samples as the initial half of the data samples for a seconddata set, which is shown in FIG. 4b as attenuated by an attenuationfactor 14. The latter half of the data samples in this second data set;that is, those samples monitored between times T-2 and Times T-3,become, in turn, the initial half of a third data set which, as shown inFIG. 40, is attenuated by the factor [6.

It can be seen from FIGS. 4a through 4c that, for a cosine squaredfunction, each data sample is utilized as an input sample for the FastFourier Computer twice, and that the data which formed the second halfof one data set will form the first half of the next succeeding dataset.

With reference to the curves 12a, 14a and 16a of FIG. 4d, it can be seenthat, if these curves are superimposed and added together, a constantunattenuated curve 18 will result. Since, as mentioned above, the outputtime domain samples which result from a Fast Fourier Transform followedby an inversed Fast Fourier Transform will show the effects of theinitial attenuation the simple superimposition of the output time domainsamples in the manner shown in FIG. 4:! will result in a set ofunattenuated time domain coefficients. Thus, for example, when thecosine squared function is used as the attenuating factor, it ispossible to produce unattenuated output samples by adding thecoefficients which form the latter half of each output data set to thecoefficients which form the initial half of samples of the nextsucceeding data set so that each final output sam ple results from theaddition of two output time domain samples from the Fast FourierTransform computer. All data is therefore redundantly calculated, whichredundant handling, by the addition techniques described, removes theeffects of the superimposed data windows. To clarify the superimpositionwhich must be performed as this case where N 8, and the attenuationcurve is a hanning curve, data samples X X X and X (FIG. 1) of one dataset will form the input data samples X,,, X X and X respectively, of thenext succeeding data set, with the data samples X,, X X and X of thatnext succeeding data set being formed by new input data. lfthese datasets are transformed using the fast Fourier algorithm, and subsequentlyinversely transformed to produce a first coefficient set havingcoefficients X through X',-, and a second coefficient set havingcoefficients X'Q, through X'}; X' X' X and X will be added to X" X" X"and X" respectively. to produce output coefficients without the effectof the initial attenuation curve.

FIGS. 5a through 5d show the attenuation curves 20, 22 and 24 for atrapezoidal window. In this instance. it can be seen that the total dataset is not redundantly handled but only the attenuated portions thereof.That is. only those data samples which occur between times T-3 and T4 ofFIGS. 50 and 5h or between times T-5 and T-6 of FIGS. 51) and 5c areredundantly handled, since no attenuation is applied to the sampleswhich occur between times T-4 and T-S of FIG. 5b, for example. It istherefore only necessary to overlap the samples at the output of theFast Fourier Transform computer by a time identical to the time betweenT-3 and T-4 to produce an unattcnuated data sample level as shown by thecurve 26 in FIG. 5d. It is therefore clear that the amount of redundancywhich is required for a particular data attenuation curve will depend onthe attenuation curve itself, but that many attenuation curves will lendthemselves to a particular percentage of redundancy which will alleviateall of the effects of the attenuation.

D. System Block Diagram Referring now to FIG. 6, a block diagram of thepreferred embodiment computer system is shown. Although the preferredembodiment conducted a L024 point transform, for ease of illustration,FIG. 6 shows an eight point Fast Fourier transform computer 28 whichaccomplishes the calculations diagrammed in FIG. I. Input data issampled in serial form from an input waveform and is applied to an inputbuffer 30. When a complete data set is available at the input buffer 30(8 data samples in this example), such as the data which appears betweentimes T-() and T-2 of FIG. 4a, the input buffer 30 is accessed inparallel to a cosine squared attenuator 32. This attenuator superimposesthe cosine squared curve such as the curve 12 of FIG. 4a on the inputdata samples. These attenuated data samples are then addressed to theFast Fourier Analysis computer 28. The output of the Fast FourierAnalysis computer, as described above, is in reverse binary order. It iscustomary when doing analysis upon sampled time domain data to conductcertain calculations on the frequency domain coefficients in theirreverse bi nary order. This is accomplished, in the preferredembodiment, by a frequency domain computer 34. An example of thecalculations which may be performed by the frequency domain computer 34is a superimposition of the frequency domain coefficients with thecharacteristics of a filter whose frequency domain characteristics areknown.

The output of the frequency domain computer 34 is then applied to aninverse fast Fourier Analysis computer 36 which converts the frequencydomain coefficients back to time domain coefficients. The output of theinverse Fast Fourier Analysis computer 36, as described above, is innormal binary order, with the initial time domain coefficient appearingon line 38 and the final time domain coefficient appearing on line 40.The

lines 42 through 52 carry the six time domain coefficients which resultfrom the intermediate input time domain coefficients monitored betweentimes T4) and T2 in FIG. 4a.

In order to superimpose the latter half of these time domaincoefficients; that is, coefficients appearing on lines 48, 50, 52 and 40with the initial half of the coefficients in the next succeeding dataset; that is, the data samples on lines 38, 42, 44 and 46, the lines 48.50, 52 and 40 are applied to a delay 54. The delay 54 delays each of theinput coefficients by the time required to access one half of a completedata set; that is, a time equal to the difference between T-l and T-2 ofFIG. 4a. The output of the delay 54 is then applied to a series ofadders 56, 58, 60 and 62 to which are likewise applied the undelayedcoefficients on the lines 38, 42, 44 and 46. These adders 58 and 62superimpose the delayed time domain coefficients from the delay 54 whichrepresent the latter half of a previous data set with the undelayedcoefficients on lines 38, 42, 44 and 46 which represent the initial halfof a succeeding data set, thus resulting in the unattenuated set ofcoefficients as shown by the curve 18 in FIG. 4d.

These output coefficients from the adders 56-62 are then applied to anoutput buffer 64 which allows serial accessing of the output data fordisplay or other use.

As will be readily recognized, the number of samples which are appliedto the delay 54 as well as the time duration of the delay 54, willdepend upon the particular attenuation function which is used. Thus, ifthe trapezoidal attenuation function shown in FIGS. 5a through 50' isutilized, the delay 54 will be applied to fewer than half of the outputtime domain data sets and will be of a duration which is equal to thedifference between times T-3 and T-4 of FIG. 50.

Using this invention, it is therefore possible, without applying aninverse attenuation curve to the calculated time domain coefficients, toremove the effects of the time domain data sampling attenuation byredundantly handling a predetermined portion of the input time domainsamples.

I claim:

I. A system for transforming input time domain data samples. comprising:

a buffer responsive to said input time domain samples for combining saidinput time domain samples into data sets, selected ones of said inputtime domain samples being combined into more than one data set;

an attenuator responsive to said data sets produced in said buffer forattenuating certain of said input time domain samples of said data setsto produce attenuated output data sets;

a first computer receiving said attenuated output data sets forperforming calculations on said output data sets according to theCooley-Tukey fast Fourier transform algorithm to produce frequencydomain coefficients;

a second computer coupled to said first computer for performingcalculations on said frequency domain coefficients according to thereverse Cooley-Tukey fast Fourier transform algorithm to produce timedomain coefficients; and

means responsive to said second computer for com bining saidcoefficients to remove from said coefficients the effects of saidattenuator.

2. A system for transforming input time domain data samples as definedin claim 1 wherein said means for removing the effects of saidattenuator additionally comprises;

means coupled to the output of said second computer for delaying aselected plurality of said time domain coefficients to produce delayedcoefficients.

3. A system for transforming input time domain data samples as definedin claim 2 wherein said means for removing the effects of saidattenuator additionally comprises a plurality of adding means, eachhaving an input from said delay means and an input from said secondcomputer, said means adding selected ones of said time domaincoefficients to selected ones of said delayed time domain coefficients.

4. A method of handling the input and output data of a fast Fourieranalysis computer, said computer capable of manipulating input timedomain data in data sets to produce output frequency domain coefficientsin data sets in accordance with a fast Fourier transform algorithm. andsaid computer capable of manipulating input frequency domaincoefficients in data sets to produce output time domain coefficients indata sets in accordance with an inverse fast Fourier transform algorithm, comprising:

accessing a series of input time domain signal samples for saidcomputer;

delaying said series of input time domain signal samples into first datasets, each data set including the data for one fast Fourier transform.each data set including a plurality of input time domain signal sampleswhich are also included in another of said data sets;

attenuating said first data sets in accordance with a predeterminedwaveform to produce attenuated data sets;

conducting said attenuating data sets to said computer after saidattenuating step as said input time domain data in data sets, so thatsaid computer will produce said output frequency domain coefficients indata sets in accordance with a fast Fourier transform algorithm;

conducting said output frequency domain coefficients in data sets tosaid computer as said input frequency domain data in data sets, so thatsaid computer will produce said output time domain co efficients in datasets in accordance with an inverse fast Fourier transform algorithm;

delaying a portion of said output time domain coefficients in each ofsaid time domain coefficient data sets resulting from said inversetransform; and

combining selected output time domain coefficients and selected ones ofsaid portion of said output time domain coefficients after said delayingstep said dividing and combining steps effecting a re moval of theeffects of said attenuation.

5. A method of handling a series of input time domain signal samples ina computer, comprising the steps of:

A. selecting a set of N of said input time domain samples;

B, superimposing an attenuation function on said set of N input timedomain signal samples to produce attenuated signal samples;

C. transforming said attenuated signal samples to produce frequencydomain coefficients from said samples in accordance with a forward fastFourier transform algorithm;

D. transforming said frequency domain coefficients to produce timedomain coefficients in accordance with an inverse fast Fourier transformalgorithm;

E. selecting an additional set of N of said input time domain samples,said additional set of N including a plurality of input time domainsamples which are also included in said set of N input time domainsamples selected in step (A);

F repeating steps ((8), (C) and (D) utilizing the set of N input timedomain samples seiected in step (E); and

G. combining selected time domain coefficients produced in step (D) withselected time domain coeffi cients produced in step (F) to remove theeffects of said attenuation function 6. A system for calculating thediscrete Fourier trans form of a plurality of serial input time domainsamples to produce output frequency domain coefficients {A(r) and forcalculating the inverse discrete Fourier transform comprising:

a buffer responsive to said serial input time domain samples for storingsaid serial input time domain samples, said buffer outputting saidserial input time domain samples in a series of data sets eachcomprising N input time domain samples, each successive data set of saidseries including a plurality of input time domain samples which are alsoincluded in the next succeeding data set in the data set series;

an attenuator connected to receive the output sets of data from saidbuffer, said attenuator superimposing an attenuation waveform upon saidsets of data from said buffer to produce a series of attenuated datasets having attenuated time domain samples {MD};

a computer responsive to said attenuator for combining said data sets ofN attenuated time domain samples in accordance with the equation toproduce a series of sets of N output frequency domain coefficients{A(r)};

means for superimposing upon said sets of N frequency domaincoefficients {Atn} a predetermined frequency characteristic to producesets of N frequency domain coefficients [A'(r)1;

a second computer responsive to said means for su perimposing forcombining said data sets of N frequency domain coefficients {A'trJ} inaccordance with the equation to produce a series of sets of N outputtime domain coefficients {X}; and

means coupled to said second computer and receiving said series of setsof N output time domain coef ficients {XX/U1 for removing the effects ofsaid attenuator from said series of sets of N output time domaincoefficients X'(k)].

domain coefficients {X"(k)} of each of said sets of N output time domaincoefficients {X'(k)}; and

means responsive to said second computer and said second buffer foradding sele ted one of said delayed time coefficients X"(k)i of each ofsaid sets of N output time domain coefficients mm} with selected ones ofsaid output time domain coefficients {X'(k)} of the next succeeding setof N output time domain coefficients {X'(k)} to remove the effect ofsaid attenuator.

1. A system for transforming input time domain data samples, comprising:a buffer responsive to said input time domain samples for combining saidinput time domain samples into data sets, selected ones of said inputtime domain samples being combined into more than one data set; anattenuator responsive to said data sets produced in said buffer forattenuating certain of said input time domain samples of said data setsto produce attenuated output data sets; a first computer receiving saidattenuated output data sets for performing calculations on said outputdata sets according to the Cooley-Tukey fast Fourier transform algorithmto produce frequency domain coefficients; a second computer coupled tosaid first computer for performing calculations on said frequency domaincoefficients according to the reverse Cooley-Tukey fast Fouriertransform algorithm to produce time domain coefficients; and meansresponsive to said second computer for combining said coefficients toremove from said coefficients the effects of said attenuator.
 2. Asystem for transforming input time domain data samples as defined inclaim 1 wherein said means for removing the effects of said attenuatoradditionally comprises: means coupled to the output of said secondcomputer for delaying a selected plurality of said time domaincoefficients to produce delayed coefficients.
 3. A system fortransforming input time domain data samples as defined in claim 2wherein said means for removing the effects of said attenuatoradditionally comprises: a plurality of adding means, each having aninput from said delay means and an input from said second computer, saidmeans adding selected ones of said time domain coefficients to selectedones of said delayed time domain coefficients.
 4. A method of handlingthe input and output data of a fast Fourier analysis computer, saidcomputer capable of manipulating input time domain data in data sets toproduce output frequency domain coefficients in data sets in accordancewith a fast Fourier transform algorithm, and said computer capable ofmaNipulating input frequency domain coefficients in data sets to produceoutput time domain coefficients in data sets in accordance with aninverse fast Fourier transform algorithm, comprising: accessing a seriesof input time domain signal samples for said computer; delaying saidseries of input time domain signal samples into first data sets, eachdata set including the data for one fast Fourier transform, each dataset including a plurality of input time domain signal samples which arealso included in another of said data sets; attenuating said first datasets in accordance with a predetermined waveform to produce attenuateddata sets; conducting said attenuating data sets to said computer aftersaid attenuating step as said input time domain data in data sets, sothat said computer will produce said output frequency domaincoefficients in data sets in accordance with a fast Fourier transformalgorithm; conducting said output frequency domain coefficients in datasets to said computer as said input frequency domain data in data sets,so that said computer will produce said output time domain coefficientsin data sets in accordance with an inverse fast Fourier transformalgorithm; delaying a portion of said output time domain coefficients ineach of said time domain coefficient data sets resulting from saidinverse transform; and combining selected output time domaincoefficients and selected ones of said portion of said output timedomain coefficients after said delaying step, said dividing andcombining steps effecting a removal of the effects of said attenuation.5. A method of handling a series of input time domain signal samples ina computer, comprising the steps of: A. selecting a set of N of saidinput time domain samples; B. superimposing an attenuation function onsaid set of N input time domain signal samples to produce attenuatedsignal samples; C. transforming said attenuated signal samples toproduce frequency domain coefficients from said samples in accordancewith a forward fast Fourier transform algorithm; D. transforming saidfrequency domain coefficients to produce time domain coefficients inaccordance with an inverse fast Fourier transform algorithm; E.selecting an additional set of N of said input time domain samples, saidadditional set of N including a plurality of input time domain sampleswhich are also included in said set of N input time domain samplesselected in step (A); F. repeating steps ((B), (C) and (D), utilizingthe set of N input time domain samples selected in step (E); and G.combining selected time domain coefficients produced in step (D) withselected time domain coefficients produced in step (F) to remove theeffects of said attenuation function.
 6. A system for calculating thediscrete Fourier transform of a plurality of serial input time domainsamples to produce output frequency domain coefficients (A(r)) and forcalculating the inverse discrete Fourier transform comprising: a bufferresponsive to said serial input time domain samples for storing saidserial input time domain samples, said buffer outputting said serialinput time domain samples in a series of data sets each comprising Ninput time domain samples, each successive data set of said seriesincluding a plurality of input time domain samples which are alsoincluded in the next succeeding data set in the data set series; anattenuator connected to receive the output sets of data from saidbuffer, said attenuator superimposing an attenuation waveform upon saidsets of data from said buffer to produce a series of attenuated datasets having attenuated time domain samples (X(k)); a computer responsiveto said attenuator for combining said data sets of N attenuated timedomain samples in accordance with the equation
 7. A system as defined inclaim 6 wherein said means for removing comprises means for combiningcertain coefficients of each of said series of sets of N coefficients(X''(k)) with certain coefficients to different sets of N coefficients(X''(k)).
 8. A system as defined in claim 6 wherein said means forremoving comprises: a second buffer responsive to said second computerfor delaying selected ones of said time domain coefficients (X''(k)) toproduce selected delayed time domain coefficients (X''''(k)) of each ofsaid sets of N output time domain coefficients (X''(k)); and meansresponsive to said second computer and said second buffer for addingselected one of said delayed time coefficients (X''''(k)) of each ofsaid sets of N output time domain coefficients (X''(k)) with selectedones of said output time domain coefficients (X''(k)) of the nextsucceeding set of N output time domain coefficients (X''(k)) to removethe effect of said attenuator.